Sharp Stability of Brenier Potentials and its Link to Poincaré Inequalities

Abstract

We prove a new functional inequality for uniformly log-concave measures, stronger than the Poincaré inequality. This new inequality bounds the variance of differences of convex functions and is motivated by the stability theory of optimal transport: for a fixed uniformly log-concave source measure, it provides a sharp stability bound for Brenier potentials with respect to variations of the target measure. Applying this inequality to Gaussian measures, we recover the fact that the Bures-Wasserstein distance between two positive semidefinite matrices is equivalent to the Frobenius distance between their square roots. Our proofs hinge upon linearization arguments for the Monge-Ampère equation governing the quadratic optimal transport problem.